Finite Group Databasebeta

← C23×#9966 #19157592

Subgroups — C23×#9966 #19157592

Order 5888.

1020 conjugacy classes of subgroups; only the 138 normal subgroups are shown.

normal subgroups only

This group has 1020 conjugacy classes of subgroups — too many to draw. Only its 138 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C23×#450C212yesmaximal
2C23×#2659C211yesmaximal, characteristic, complemented
3C2×C23×#328C212yesmaximal, complemented
4C23×#1438C211yesmaximal, characteristic
5C23×#1280C211yesmaximal, characteristic, complemented
6C23×D4×C23C2211yescharacteristic
7C22×SD16×C23—22no—
8C2×C23×C22⋊Q8C2212yes—
9C22×C22⋊C4×C23—22no—
10C23×#450—24no—
11C23×#450—24no—
12C2×C23×#102—22no—
13C2×C23×C22⋊C8C2212yes—
14C23×#398—22no—
15C23×#395—22no—
16C23×#380—22no—
17C2×C23×#95C2211yescharacteristic
18C2×C23×#95C2211yescharacteristic
19C23×#328—24no—
20C25×C23—21no—
21C22×Q8×C23D412yes—
22C22×D4×C23—22no—
23C22×D4×C23—22no—
24C22×D4×C23—21no—
25C22×D4×C23—44no—
26C22×D4×C23D412yes—
27C23×C4×C23—21no—
28C23×C4×C23D412yes—
29C23×C4×C23C2311yescharacteristic
30C2×SD16×C23—28no—
31C2×SD16×C23—48no—
32C22×C8×C23—22no—
33C23×C22⋊Q8—22no—
34C23×C22⋊Q8—22no—
35C23×C22⋊Q8—22no—
36C23×C22⋊Q8—22no—
37C2×C4⋊C4×C23—22no—
38C2×C4⋊C4×C23—22no—
39C2×C4⋊C4×C23—22no—
40C2×C4⋊C4×C23D412yes—
41C2×C22⋊C4×C23—22no—
42C2×C22⋊C4×C23—44no—
43C2×C22⋊C4×C23—44no—
44C2×C22⋊C4×C23—44no—
45C23×#102—28no—
46C23×C22⋊C8—22no—
47C23×C22⋊C8D414yes—
48C23×#95—24no—
49C23×#95—22no—
50C23×#95—22no—
51C24×C23—24no—
52C24×C23—22no—
53C24×C23—42no—
54C24×C23—42no—
55C24×C23—42no—
56C24×C23—41no—
57C24×C23—41no—
58C24×C23—42no—
59C24×C23—42no—
60C24×C23C2×D411yescharacteristic
61C2×Q8×C23—22no—
62C2×Q8×C23—22no—
63C2×Q8×C23—44no—
64C2×D4×C23—28no—
65C2×D4×C23—24no—
66C2×D4×C23—44no—
67C2×D4×C23—44no—
68C2×D4×C23—48no—
69C2×D4×C23—44no—
70C2×D4×C23—42no—
71C22×C4×C23—22no—
72C22×C4×C23—22no—
73C22×C4×C23—22no—
74C22×C4×C23—22no—
75C22×C4×C23—22no—
76C22×C4×C23—42no—
77C22×C4×C23—42no—
78C22×C4×C23—42no—
79C22×C4×C23—42no—
80C22×C4×C23—41no—
81C22×C4×C23—41no—
82C22×C4×C23—42no—
83C22×C4×C23SD1614yescomplemented
84C22×C4×C23C2×D412yes—
85C22×C4×C23C2×D411yescharacteristic
86C22×C4×C23C2×D412yes—
87C22×C4×C23#4111yescharacteristic
88SD16×C23—416no—
89C2×C8×C23—24no—
90C2×C8×C23—44no—
91C4⋊C4×C23—22no—
92C4⋊C4×C23—22no—
93C4⋊C4×C23—42no—
94C4⋊C4×C23—42no—
95C4⋊C4×C23—82no—
96C4⋊C4×C23—82no—
97C22⋊C4×C23—416no—
98C22⋊C4×C23—42no—
99C22⋊C4×C23—42no—
100#9966C2311yesmaximal, characteristic, direct product factor, Sylow
101C23×C23—21no—
102C23×C23—21no—
103C23×C23—22no—
104C23×C23—24no—
105C23×C23—44no—
106C23×C23—44no—
107C23×C23—42no—
108C23×C23—44no—
109C23×C23—44no—
110C23×C23—44no—
111C23×C23—44no—
112C23×C23—42no—
113C23×C23—44no—
114C23×C23—44no—
115C23×C23—82no—
116C23×C23—82no—
117C23×C23—82no—
118C23×C23—82no—
119C23×C23—82no—
120C23×C23—82no—
121C23×C23—81no—
122C23×C23—81no—
123C23×C23—81no—
124C23×C23—81no—
125C23×C23#12011yescharacteristic
126C23×C23#12111yescharacteristic
127C23×C23#12711yescharacteristic
128C23×C23C2×SD1612yes—
129C23×C23#13712yes—
130Q8×C23—44no—
131Q8×C23—82no—
132D4×C23—28no—
133D4×C23—44no—
134D4×C23—48no—
135C2×C4×C23—24no—
136C2×C4×C23—24no—
137C2×C4×C23—41no—
138C2×C4×C23—42no—
139C2×C4×C23—41no—
140C2×C4×C23—42no—
141C2×C4×C23—42no—
142C2×C4×C23—44no—
143C2×C4×C23—44no—
144C2×C4×C23—42no—
145C2×C4×C23—42no—
146C2×C4×C23—42no—
147C2×C4×C23—42no—
148C2×C4×C23—42no—
149C2×C4×C23—44no—
150C2×C4×C23—82no—
151C2×C4×C23—82no—
152C2×C4×C23—81no—
153C2×C4×C23—81no—
154C2×C4×C23#12012yes—
155C2×C4×C23#12112yes—
156C8×C23—44no—
157C2×#450C2×C2312yes—
158#2659C2×C2311yescharacteristic, complemented
159C2×#328C2×C2312yescomplemented
160#1438C2×C2311yescharacteristic
161#1280C2×C2311yescharacteristic, complemented
162C22×C23—22no—
163C22×C23—22no—
164C22×C23—22no—
165C22×C23—22no—
166C22×C23—21no—
167C22×C23—21no—
168C22×C23—21no—
169C22×C23—21no—
170C22×C23—48no—
171C22×C23—44no—
172C22×C23—84no—
173C22×C23—84no—
174C22×C23—84no—
175C22×C23—84no—
176C22×C23—84no—
177C22×C23—82no—
178C22×C23—82no—
179C22×C23#39211yescharacteristic
180C22×C23#44812yes—
181C22×C23#45311yescharacteristic
182C22×C23#45411yescharacteristic
183C22×C23B(4,𝔽2)12yes—
184C22×C23#46112yes—
185C22×C23#49211yescharacteristic
186C22×C23#49711yescharacteristic
187C4×C23—22no—
188C4×C23—22no—
189C4×C23—44no—
190C4×C23—82no—
191C4×C23—82no—
192C4×C23—82no—
193C23×D4C22×C2311yescharacteristic
194C22×SD16—22no—
195C2×C22⋊Q8C22×C2312yes—
196C22×C22⋊C4—22no—
197#450—24no—
198#450—24no—
199C2×#102—22no—
200C2×C22⋊C8C22×C2312yes—
201#398—22no—
202#395—22no—
203#380—22no—
204C2×#95C22×C2311yescharacteristic
205C2×#95C22×C2311yescharacteristic
206#328—24no—
207C2×C23—21no—
208C2×C23—22no—
209C2×C23—21no—
210C2×C23—84no—
211C2×C23#159612yes—
212C2×C23#159912yes—
213C2×C23#200611yescharacteristic
214C2×C23#201311yescharacteristic
215C2×C23#202111yescharacteristic
216C25—21no—
217C22×Q8D4×C2312yes—
218C22×D4—22no—
219C22×D4—22no—
220C22×D4—21no—
221C22×D4—44no—
222C22×D4D4×C2312yes—
223C23×C4—21no—
224C23×C4D4×C2312yes—
225C23×C4C23×C2311yescharacteristic
226C2×SD16—28no—
227C2×SD16—48no—
228C22×C8—22no—
229C22⋊Q8—22no—
230C22⋊Q8—22no—
231C22⋊Q8—22no—
232C22⋊Q8—22no—
233C2×C4⋊C4—22no—
234C2×C4⋊C4—22no—
235C2×C4⋊C4—22no—
236C2×C4⋊C4D4×C2312yes—
237C2×C22⋊C4—22no—
238C2×C22⋊C4—44no—
239C2×C22⋊C4—44no—
240C2×C22⋊C4—44no—
241#102—28no—
242C22⋊C8—22no—
243C22⋊C8D4×C2314yes—
244#95—24no—
245#95—22no—
246#95—22no—
247C23#996611yescharacteristic, direct product factor, Sylow
248C24—24no—
249C24—22no—
250C24—42no—
251C24—42no—
252C24—42no—
253C24—41no—
254C24—41no—
255C24—42no—
256C24—42no—
257C24C2×D4×C2311yescharacteristic
258C2×Q8—22no—
259C2×Q8—22no—
260C2×Q8—44no—
261C2×D4—28no—
262C2×D4—24no—
263C2×D4—44no—
264C2×D4—44no—
265C2×D4—48no—
266C2×D4—44no—
267C2×D4—42no—
268C22×C4—22no—
269C22×C4—22no—
270C22×C4—22no—
271C22×C4—22no—
272C22×C4—22no—
273C22×C4—42no—
274C22×C4—42no—
275C22×C4—42no—
276C22×C4—42no—
277C22×C4—41no—
278C22×C4—41no—
279C22×C4—42no—
280C22×C4SD16×C2314yescomplemented
281C22×C4C2×D4×C2312yes—
282C22×C4C2×D4×C2311yescharacteristic
283C22×C4C2×D4×C2312yes—
284C22×C4#41×C2311yescharacteristic
285SD16—416no—
286C2×C8—24no—
287C2×C8—44no—
288C4⋊C4—22no—
289C4⋊C4—22no—
290C4⋊C4—42no—
291C4⋊C4—42no—
292C4⋊C4—82no—
293C4⋊C4—82no—
294C22⋊C4—416no—
295C22⋊C4—42no—
296C22⋊C4—42no—
297C23—21no—
298C23—21no—
299C23—22no—
300C23—24no—
301C23—44no—
302C23—44no—
303C23—42no—
304C23—44no—
305C23—44no—
306C23—44no—
307C23—44no—
308C23—42no—
309C23—44no—
310C23—44no—
311C23—82no—
312C23—82no—
313C23—82no—
314C23—82no—
315C23—82no—
316C23—82no—
317C23—81no—
318C23—81no—
319C23—81no—
320C23—81no—
321C23C23×#12011yescharacteristic
322C23C23×#12111yescharacteristic
323C23C23×#12711yescharacteristic
324C23C2×SD16×C2312yes—
325C23C23×#13712yes—
326Q8—44no—
327Q8—82no—
328D4—28no—
329D4—44no—
330D4—48no—
331C2×C4—24no—
332C2×C4—24no—
333C2×C4—41no—
334C2×C4—42no—
335C2×C4—41no—
336C2×C4—42no—
337C2×C4—42no—
338C2×C4—44no—
339C2×C4—44no—
340C2×C4—42no—
341C2×C4—42no—
342C2×C4—42no—
343C2×C4—42no—
344C2×C4—42no—
345C2×C4—44no—
346C2×C4—82no—
347C2×C4—81no—
348C2×C4—82no—
349C2×C4—81no—
350C2×C4C23×#12012yes—
351C2×C4C23×#12112yes—
352C8—44no—
353C22—21no—
354C22—21no—
355C22—21no—
356C22—21no—
357C22—22no—
358C22—22no—
359C22—22no—
360C22—22no—
361C22—48no—
362C22—44no—
363C22—84no—
364C22—84no—
365C22—84no—
366C22—82no—
367C22—82no—
368C22—84no—
369C22—84no—
370C22C23×#39211yescharacteristic
371C22C23×#44812yes—
372C22C23×#45311yescharacteristic
373C22C23×#45411yescharacteristic
374C22C23×B(4,𝔽2)12yes—
375C22C23×#46112yes—
376C22C23×#49211yescharacteristic
377C22C23×#49711yescharacteristic
378C4—22no—
379C4—22no—
380C4—44no—
381C4—82no—
382C4—82no—
383C4—82no—
384C2—21no—
385C2—22no—
386C2—21no—
387C2—84no—
388C2C23×#159612yes—
389C2C23×#159912yes—
390C2C23×#200611yescharacteristic
391C2C23×#201311yescharacteristic
392C2C23×#202111yescharacteristic
subgroupG/H
center, socleC23×C23#120
commutator, Frattini, cosocleC23×C4C23×C23
prime radicalC23×D4×C23C22
LCSG ▷ C23×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23×C23 ◁ C23×C4×C23 ◁ G

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. C23×#2659C2semidirect
3. C2×C23×#328C2semidirect
5. C23×#1280C2semidirect
83. C22×C4×C23SD16semidirect
100. #9966C23direct (N × H)
158. #2659C2×C23semidirect
159. C2×#328C2×C23semidirect
161. #1280C2×C23semidirect
247. C23#9966direct (N × H)
280. C22×C4SD16×C23semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C2×C23×#4502
C23×#26592
C2×C23×#3282
C23×#14382
C23×#12802
#996623